On highly accurate computations of the Coulomb two-center systems with unit charges

Results of our recent highly accurate computations of the Coulomb two-center systems with the unit electrical charges $X^{+} X^{+} e^{-}$ and $X^{+} Y^{+} e^{-}$ are discussed. In particular, we have determined (to very high accuracy) the total energies of the ground $1 s \sigma-$states in the two-center adiabatic (or molecular) H$_{2}^{+}$, D$_2^{+}$, HD$^{+}$, HT$^{+}$, T$^{+}_{2}$ and DT$^{+}$ ions. In these computations we applied the new masses of hydrogen isotopes, which have been measured in the recent high-energy experiments. We also derived (and tested) our accurate mass-interpolation formula for the total energies of the model two-center $X^{+} X^{+} e^{-}$ ions with very heavy masses of the point $X^{+}$ particles ($M_X \ge$ 100,000 $m_e$). By using this formula we analyze the overall accuracy and validity of the adiabatic two-center approximation in application to the three-body atomic (or Coulomb) systems.